Melles Griot Optics Guide

Performance Factors

Lens Shape



Aberrations described in the preceding section are highly dependent on application, lens shape, and material of the lens (or, more exactly, its index of refraction). The singlet shape that minimizes spherical aberration at a given conjugate ratio is called best-form. The criterion for best-form at any conjugate ratio is that the marginal rays are equally refracted at each of the lens/air interfaces. This minimizes the effect of sinq not equal to q. It is also the criterion for minimum surface-reflectance loss. Another benefit is that absolute coma is nearly minimized for best-form shape, at both infinite and unit conjugate ratios. To further explore the dependence of aberrations on lens shape, it is helpful to make use of the Coddington shape factor, q, defined as

The figure below shows the transverse and longitudinal spherical aberration of a singlet lens as a function of the shape factor, q. In this particular instance, the lens has a focal length of 100 mm, operates at f/# = 5, has an index of refraction of 1.518722 (BK7 at the mercury green line, 546.1 nm), and is being operated at the infinite conjugate ratio. It is also assumed that the lens itself is the aperture stop. An asymmetric shape that corresponds to a q-value of about 0.7426 for this material and wavelength is the best singlet shape for on-axis imaging. Best-form shapes are used in Melles Griot laser-line-focusing singlet lenses. It is important to note that the best-form shape is dependent on refractive index. For example, with a high-index material, such as silicon, the best-form lens for the infinite conjugate ratio is a meniscus shape.

Aberrations of positive singlets as a function of shape for infinite conjugate ratio
At infinite conjugate with a typical glass singlet, the plano-convex shape (q = 1), with convex side toward the infinite conjugate, performs nearly as well as the best-form lens. Because a plano-convex lens costs much less to manufacture than an asymmetric biconvex singlet, these lenses are quite popular. Furthermore, this lens shape exhibits near-minimum total transverse aberration and near-zero coma when used off axis, thus enhancing its utility. For imaging at unit magnification (s = s" = 2f ), a similar analysis would show that a symmetric biconvex lens is the best shape. Not only is spherical aberration minimized, but coma, distortion, and lateral chromatic aberration exactly cancel each other out. These results are true regardless of material index or wavelength, which explains the utility of symmetric convex lenses, as well as symmetrical optical systems in general. However, if a remote stop is present, these aberrations may not cancel each other quite as well. For wide-field applications, the best-form shape is definitely not the optimum singlet shape, especially at the infinite conjugate ratio, since it yields maximum field curvature. The ideal shape is determined by the situation and may require rigorous ray-tracing analysis. It is possible to achieve much better correction in an optical system by using more than one element. The cases of an infinite conjugate ratio system and a unit conjugate ratio system are discussed in the following section.

Previous Next